cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322190 E.g.f.: A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n!*k!), as a square table of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 4, 1, 1, 8, 14, 14, 8, 1, 1, 16, 41, 52, 41, 16, 1, 1, 32, 122, 200, 200, 122, 32, 1, 1, 64, 365, 784, 977, 784, 365, 64, 1, 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1, 1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1, 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1, 1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1, 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2018

Keywords

Comments

Compare to the addition theorem of Jacobi's elliptic functions: cn(x+y) + i*sn(x+y) = (cn(x) + i*sn(x)*dn(y)) * (cn(y) + i*sn(y)*dn(x)) / (1 - k^2*sn(x)^2*sn(y)^2), where the modulus k is implicit.
See A322620 for another description of the e.g.f. of this sequence.

Examples

			E.g.f.: A(x,y) = 1 + (1*x + 1*y) + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
where A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
This square table of coefficients of x^n*y^k/(n!*k!) in A(x,y) begins
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...;
1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, ...;
1, 4, 14, 52, 200, 784, 3104, 12352, 49280, 196864, ...;
1, 8, 41, 200, 977, 4808, 23801, 118280, 589217, 2939528, ...;
1, 16, 122, 784, 4808, 29056, 174752, 1049344, 6297728, 37789696, ...;
1, 32, 365, 3104, 23801, 174752, 1257125, 8948384, 63318641, 446442272, ...;
1, 64, 1094, 12352, 118280, 1049344, 8948384, 74628352, 614111360, 5010663424, ...;
1, 128, 3281, 49280, 589217, 6297728, 63318641, 614111360, 5823720257, 54420050048, ...; ...
This sequence may be written as a triangle, starting as
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 4, 5, 4, 1;
1, 8, 14, 14, 8, 1;
1, 16, 41, 52, 41, 16, 1;
1, 32, 122, 200, 200, 122, 32, 1;
1, 64, 365, 784, 977, 784, 365, 64, 1;
1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;
1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1;
1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1; ...
RELATED SERIES.
The series expansions for C(x,y) and S(x,y) are given by
C(x,y) = 1 + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
S(x,y) = (1*x + 1*y) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + ...
where A(x,y) = C(x,y) + S(x,y) such that C(x,y)^2 - S(x,y)^2 = 1.
The e.g.f. may be written with coefficients of x^n*y^k/(n+k)!, as follows:
A(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^6 + 96*x^5*y + 615*x^4*y^2 + 1040*x^3*y^3 + 615*x^2*y^4 + 96*x*y^5 + 1*y^6)/6! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + (1*x^8 + 512*x^7*y + 10220*x^6*y^2 + 43904*x^5*y^3 + 68390*x^4*y^4 + 43904*x^3*y^5 + 10220*x^2*y^6 + 512*x*y^7 + 1*y^8)/8! + ...
these coefficients are described by table A322620.

Links

Crossrefs

Cf. A322193 (C(x,y)), A322194 (S(x,y)), A322195 (main diagonal), A322196, A322197.
Cf. A322620, A245140, A245155, A245166.

Programs

  • Mathematica
    nmax = 13;
t[n_, k_] := SeriesCoefficient[(Cosh[x] Cosh[y] + Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, n}, {y, 0, k}] n! k!;
Table[t[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 26 2018 *)
  • PARI
    {T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
n!*k!*polcoeff(polcoeff(C + S,n,x),k,y)}
/* Print as a square table */
for(n=0,10,for(k=0,10,print1( T(n,k),", "));print(""))
/* Print as a triangle */
for(n=0,15,for(k=0,n,print1( T(n-k,k),", "));print(""))

Formula

E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).
E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
E.g.f.: A(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
E.g.f.: A(x,y) = C(x,y) + S(x,y) such that the following identities hold.
(1) C(x,y)^2 - S(x,y)^2 = 1.
(2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)).
(2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
(3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)).
(3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)).
(3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(5a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) * (cosh(y) - sinh(y)*C(x,y)) / (1 - sinh(y)^2*S(x,y)^2).
(5b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) * (cosh(x) - sinh(x)*C(x,y)) / (1 - sinh(x)^2*S(x,y)^2).
(6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)).
(6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)).
SPECIAL ARGUMENTS.
A(x, y=0) = exp(x).
A(x, y=x) = (1 + sinh(x)) / (1 - sinh(x)).
A(x, y=-x) = 1.
FORMULAS FOR TERMS.
a(n) = A322620(n,k) / binomial(n,k).
Sum_{k=0..n} 2^k * binomial(n,k) * T(n,k) = A245140(n).
Sum_{k=0..n} 3^k * binomial(n,k) * T(n,k) = A245155(n).
Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * T(n,k) = A245166(n).

A322194 E.g.f.: S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where S(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^(2*n+1-k)*y^k/((2*n+1-k)!*k!), as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 8, 14, 14, 8, 1, 1, 32, 122, 200, 200, 122, 32, 1, 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1, 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1, 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1, 1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2018

Keywords

Comments

See A322622 for another description of the e.g.f. of this sequence.

Examples

			E.g.f.: S(x,y) = (1*x + 1*y) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + ...
where S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
This irregular triangle of coefficients of x^(2*n+1-k)*y^k/((2*n+1-k)!*k!) in C(x,y) begins
1, 1;
1, 2, 2, 1;
1, 8, 14, 14, 8, 1;
1, 32, 122, 200, 200, 122, 32, 1;
1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;
1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1;
1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1;
1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1; ...
RELATED SERIES.
The series C(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
C(x,y) = 1 + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/(2*n+1)!, as follows:
S(x,y) = (1*x + 1*y) + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + ...
these coefficients are described by triangle A322622.

Links

Crossrefs

Cf. A322190 (C + S), A322193 (C), A322196 (main diagonal).
Cf. A322622.

Programs

  • Mathematica
    T[n_, k_] := (2n-k+1)! k! SeriesCoefficient[(Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, 2n-k+1}, {y, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, 2n+1}] // Flatten (* Jean-François Alcover, Dec 29 2018 *)
  • PARI
    {T(n, k) = my(X=x+x*O(x^(2*n+1-k)), Y=y+y*O(y^k));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
(2*n+1-k)!*k!*polcoeff(polcoeff(S, 2*n+1-k, x), k, y)}
/* Print as a triangle */
for(n=0, 10, for(k=0, 2*n+1, print1( T(n, k), ", ")); print(""))

Formula

E.g.f.: S(x,y) and related series C(x,y) satisfy the following identities.
(1) C(x,y)^2 - S(x,y)^2 = 1.
(2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)).
(2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
(3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)).
(3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)).
(3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(5a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) * (cosh(y) - sinh(y)*C(x,y)) / (1 - sinh(y)^2*S(x,y)^2).
(5b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) * (cosh(x) - sinh(x)*C(x,y)) / (1 - sinh(x)^2*S(x,y)^2).
(5c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).
(6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)).
(6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)).
(6c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
(6d) C(x,y) + S(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
SPECIAL ARGUMENTS.
S(x, y=0) = sinh(x).
S(x, y=x) = 2*sinh(x) / (1 - sinh(x)^2).
S(x, y=-x) = 0.

A322625 a(n) = [x^(n+1)*y^n/(2*n+1)!] (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), for n >= 0.

Original entry on oeis.org

1, 6, 140, 7000, 605808, 80735424, 15355426944, 3951806601600, 1322951416666880, 558885077673235456, 290849071253404145664, 182840964355984767938560, 136612441159156704715366400, 119668670961667681489160601600, 121471774394099416962618586890240, 141448233929575736228827518568857600, 187296050462489978166693915859822510080
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2019

Keywords

Comments

a(n) = A322620(n+1, n) for n >= 0.
a(n) = binomial(2*n+1,n) * A322196(n) for n >= 0.

Crossrefs

Cf. A322620, A322196.

Programs

  • PARI
    {A322620(n, k) = my(X=x+x*O(x^n), Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
(n+k)!*polcoeff(polcoeff(C + S, n, x), k, y)}
for(n=0, 20, print1( A322620(n+1, n), ", "))
Showing 1-3 of 3 results.

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